Sparsifying subband decompositions

We present a solution for constructing over-complete sparse subband decompositions. This is a generalization of the perturbed basis pursuit problem (Chen, S. and Donoho, D.L., SIAM J. Sci. Computation, vol.20, no.1, p.33-61, 1999) specifically applied to an over-complete subband representation. Our formulation is based upon the iterative re-weighted least squares algorithm and can be given a probabilistic interpretation. Although the convergence properties of this algorithm are known to be slow, we observe experimentally that only a few iterations are sufficient to generate a reasonably sparse approximation. Furthermore, using subband bases provides us with an algorithm whose complexity grows linearly in time.